Congrunces of partitions into distinct parts

Question

Let $P_{d}(n)$ denote the number of partitions of n into distinct parts. The generating function of $P_{d}(n)$ s given by: $$ \sum_{n \geq 0}P_{d}(n)q^{n}= \prod_{n \geq 0} (1+q^{n}).$$ Now let $P_{2,d}(n)$ denote the number of partitions of n into distinct parts with two different colors, where the generating function is given by $$ \sum_{n \geq 0}P_{2,d}(n)q^{n}= \prod_{n \geq 0} (1+2q^{n}).$$ My questions are: how to extract Ramanujan-like congruences (or families) for $P_{2,d}(n)$ since it is obvious that we can not use the dissections of generating functions on $\prod_{n \geq 0} (1+2q^{n})$.
Also, is it possible to establish a recurrence formula for $P_{2,d}(n)$ such as the one of $P_{d}(n)$?, in which $$ P_{d}(n)= s(n)+2\sum_{k=1}^{\sqrt{n}}(-1)^{k+1}P_{d}(n-k^{2}).$$ where $ s(n)= (-1)^{j}$ if $n=j(3j+1)/2$ or $n=j(3j-1)/2$ , and $0$ otherwise.